Updating $\bar m_{c,b}(\bar m_{c,b})$ from SVZ-moments and their ratios

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Updating ¯

m_c, b( ¯

m_c, b) from SVZ-moments and their

ratios

Stephan Narison

To cite this version:

Stephan Narison. Updating ¯

m_c, b( ¯

m_c, b) from SVZ-moments and their ratios. Physics Letters B,

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Updating

m

_c

_,

_b

(

m

_c

_,

_b

)

from

SVZ-moments

and

their

ratios

Stephan Narison

LaboratoireUniversetParticules,CNRS-IN2P3,Case 070,PlaceEugèneBataillon,34095,MontpellierCedex 05,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received30May2018

Receivedinrevisedform12July2018 Accepted3August2018

Availableonline6August2018 Editor:A.Ringwald

Keywords:

QCDspectralsumrules Perturbativeandnon-perturbative calculations

Heavyquarkmasses Gluoncondensates

Using recent values of

α

s, the gluon condensates 

α

sG2 and g3fabcG3 and the new data onthe

ψ/ϒ-families, we update our determinations of the M S running quark masses mc,b(mc,b) from the

SVZ-moments _Mn(Q2) and their ratios [1,2] by including higherorder perturbative(PT) corrections,

non-perturbative(NPT)termsuptodimensiond=8 andusingthedegreen-stabilitycriteriaofthe(ratios of)moments.Optimalresultsfromdifferent(ratiosof)momentsconvergetotheaccuratemeanvalues:

mc(mc)=1264(6)MeV andmb(mb)=4188(8)MeV inTable 4,whichimproveandconfirmourprevious

findings [1,2] andtherecentonesfromLaplacesumrules [3].Commentsonsomeotherdeterminations ofmc(mc)and

α

sG2fromtheSVZ-(ratiosof)momentsinthevectorchannelaregiveninSection5.

©2018TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. IntroductionandSVZ-moments

In Refs. [1,2], we have used different

M

n

(Q

2

)

moments and

theirratiosrn

_/r

n+j_{introducedbySVZ [4,5]}1_{forextractingthe} val-uesofthecharmandbottomrunningquarkmassesmc,b

(m

c,b

)

and

thedimension4:



α

sG2



and6:



g3fabcG3



gluoncondensates.

Us-ingtherecentvaluesofthe gluoncondensatesfromLaplacesum rules [3,15] and newdata on the

ψ/ϒ

-families massesand lep-tonic widths [16], we shall improve in this paper our previous results for the quark masses. Here, we shall be concerned with thetwo-pointcorrelator:

−



gμνq2

−

qμqν







(

q2

)

≡

i



d4x e−iqx



0

|

T

Jμ_

(

x

)



Jν_

(0)



†

|

0

,

(1) associatedtothe Jμ_

= ¯

γ

μ

_

₍

_

_≡

_{c,b)heavyquarkneutral} vec-torcurrent.Thecorrespondingmomentsare2_:

E-mailaddress:snarison@yahoo.fr.

1 _{Forreviews,seee.g. [6–14].}

2 _{Weshallusethesamenormalizationas[17] andsomeoftheexpressionsgiven}

there.

M

n



−

q2

≡

Q2



≡

4

π

2

(

−

1) n n

!



d d Q2



n

(

−

Q2

)

=

∞



4m2 Q dt R

(

t

,

m 2 c

)

(

t

+

Q2

₎

n+1

.

(2)

Theirratiosread:

rn/n+1

(

Q2

)

=

M

n

(

Q2

)

M

n+1

(

Q2

)

,

rn/n+2

(

Q2

)

=

M

n

(

Q2

)

M

n+2

(

Q2

)

,

(3) where the experimental sidesare more precise than that of the moments

M

n

(Q

2

)

. It has been noticed by [18,19] thatthe OPE

of

M

n

(

0

)

breaks down for highervalues of n, while it has also

been mentioned in [1,2] that low moments n

≤

3 are sensitive tothewayforparametrizingthehigh-energypartofthespectral function(hereaftercalledQCDcontinuum)makingtheresults ob-tainedfromlowmomentsmodel-dependent.Therefore,oneshould lookforcompromise valuesofn (stabilityinn)whereboth prob-lems are avoided. Another wayout is to work with the Q2

=

0 moments [11] wheretheOPEconvergesfasterwhiletheQCD con-tinuumcontributionsarestronglysuppressed.

2. ExpressionsoftheSVZ-moments

M

n

(

Q2

)

TheQCD expressionsofthemomentscanbederived fromthe onesofR.Theon-shellexpressionofthespectralfunctionis trans-formedintotheM S-schemebyusingtheknownrelationbetween https://doi.org/10.1016/j.physletb.2018.08.003

262 S. Narison / Physics Letters B 784 (2018) 261–265 Table 1

QCDparameters.

Dimension d Name Values [GeVd_{] Refs.}

0 αs(MZ) 0.1182(19) [3,16,20–22] 4 αsG2 (6.35±0.35)10−2 [3] 6 g3_f abcG3 (8.2±1.0)GeV2αsG2 [15] 8 G4_ (0.75±025)G2_2 _[19,23] Table 2

Massesandelectronicwidthsofthe J/ψfamilyfromPDG16[16]. Name Mass [MeV] J/ψ→e+e−[keV]

J/ψ(1S) 3096.916(11) 5.55(14) ψ(2S) 3686.097(25) 2.34(4) ψ(3770) 3773.13(0.35) 0.262(18) ψ(4040) 4039(1) 0.86(7) ψ(4160) 4191(5) 0.48(22) ψ(4415) 4421(4) 0.58(7)

the on-shelland M S-scheme runningquark masses. The sources ofdifferentPT contributionsuptoorder

α

3

s for

M

n

(Q

2

=

0

)

and

up to order

α

2

s for

M

n

(Q

2

=

0

)

are quoted in [1] and will not

be re-quoted here. The same for the different NP contributions up to dimensiond

=

8 whereone notice that thed

=

4 conden-sate contributionisknown to NLO.Some explicit numericalQCD expressions of the moments can be found in Ref. [1]. We shall usetheQCDparametersgiveninTable1.Tothevalueof

α

s

(M

Z

)

quotedthere,correspond:

α

s

(

mc

)

=

0.397(15) and

α

s

(

mb

)

=

0.227(7) , (4)

where we have used the recent determinations from a recent globalfitofthe(axial-)vectorand(pseudo)scalarcharmoniumand bottomiumsystemsusingLaplacesumrules [3]:

mc

(

mc

)

=

1264(10)MeV

,

mb

(

mb

)

=

4.184(9)MeV. (5)

Thelow-energy partofthespectral functioniswell described by the sum of different resonances contributions within a narrow widthapproximation(NWA).Forthec-quarkchannel,itreads:

Rc

(

t

)

≡

4

π

Imc

(

t

+

i



)

=

π

Nc Qc2

α

2

J/ψ Mψ

ψ→e+e−

δ



t

−

M2_ψ



,

(6) whereNc

=

3; Mψ and

ψ→e+e− arethemassandleptonicwidth

ofthe J/ψmesons; Qc

=

2

/

3 isthecharmelectricchargeinunits

of e;

α

=

1

/

133

.

6 is the running electromagnetic coupling eval-uated at M2

ψ. We shall use the experimental values of the J/ψ

parameterscompiledinTable2.

Weshallparametrizethecontributionsfrom

√

tc

≥ (

4

.

5

±

0

.

1

)

GeVusingeither:

–Model1: TheapproximatePTQCD expressionofthespectral function to order

α

2

s up to order

(m

2c

/t)

6 given in [24] and the

α

3

s contributionfromnon-singletcontributionuptoorder

(m

c2

/t)

2

givenin[25].

–Model2: The asymptoticPT expressionof thespectral func-tion known to order

α

3

s where the quark mass corrections are

neglected.3

–Model3: Fitsofdifferentdataabovethe

ψ(

2S

)

mass:weshall takee.g.the resultsin[25] whereacomparisonofresultsfrom dif-ferentfittingprocedurescanbefoundinthispaper(seee.g. [26]).

3 _{OriginalpapersaregiveninRefs.317to321ofthebookinRef. [7].}

Fig. 1. Valuesofmc(mc)fromMn(0)fordifferentvaluesofn usingtheQCDinput

parametersinTable1andthethreemodelsgivenpreviouslyfortheQCDcontinuum parametrization.

Fig. 2. Valuesofmc(mc)fromtheratiosofmomentsrn/n+1(0)andrn/n+2(0)for

dif-ferentvaluesofn usingtheQCDinputparametersinTable1andModel1given previously for theQCD continuum parametrization.In then axis: 1≡r1/2,2≡ r2/3,3≡r2/4,4≡r3/4,5≡r3/5,6≡r4/5.

3. Runningmc

(

mc

)

charmquarkmassfrom

M

n

(

0

)

– Using the previous models for parametrizing the QCD con-tinuum,we show inFig.1 thevaluesofmc

(m

c

)

from

M

n

(

0

)

for

differentvaluesofn.WehaveusedtheMathematicaprogramFind Rootforextractingthevaluesofmc

(m

c

)

leftasafreeparameterin

theOPEincluding1

/m

8_c corrections.

–Onecanseethatthemodel-dependenceoftheresults disap-pear forn

≥

3 wherestabilityinn isobtained.NotingthatModel 1 gives themost conservativeresult andappears (apriori) to be a good approximationofthe spectral functionasit includeshigher orderradiative

⊕

masscorrections,weshallonlyconsiderModel 1 intherestofthepaper.Atthestabilitypointn



3

−

4,wededuce theoptimalestimate(inunitsofMeV):

mc

(

mc

)

|

40

=

1266(8.8)ex

(0.7)

αs

(5.2)

α4

s

(0.1)

G2

(0.3)

G3

(1.5)

G4

.

(7)

–Wedoasimilaranalysisfortheratiosofmomentsrn/n+1

(

0

)

andrn/n+2

(

0

)

.Theresultsversusthedegreeofmomentsareshown inFig.2.Wededuce,atthestability pointn



4,thevalue(inunits ofMeV):

mc

(

mc

)

|

03/4

=

1264(0.1)ex

(2.7)

αs

(9.9)

α4

s

(0.3)

G2

(0.2)

G3

(4.3)

G4

,

(8) whereonecannoticethattheexperimentalerrorisreduced com-paredtothe momentresultswhiletheonesinduced bytheQCD parametershaveincreased.

–Theerrorsfromthe

α

4

s-termisassumedtobeaboutthesize

ofthecontributionfromtheknown

α

3

s termwhichisagenerous

Fig. 3. Valuesofmc(mc)fromthemomentsMn(4mc2)andtheirratiosrn/n+1(4m2c)

andrn/n+2(4m2c)fordifferentvaluesofn usingtheQCDinputparametersinTable1

andModel1givenpreviouslyfortheQCDcontinuumparametrization.Inthen axis:

7≡r7/8,8≡r7/9,9≡r8/9,10≡r8/10,11≡r9/10,12≡r9/11,13≡r10/11.

Fig. 4. Valuesofmc(mc)fromthemomentsMn(8mc2)andtheirratiosrn/n+1(8m2c)

andrn/n+2(8m2c)fordifferentvaluesofn usingtheQCDinputparametersinTable1

andModel1givenpreviouslyfortheQCDcontinuumparametrization.Inthen axis:

14≡r14/16,15≡r15/16,16≡r15/17,17≡r16/17,18≡r16/18,19≡r17/18.

4. Runningmc

(

mc

)

charmquarkmassfrom

M

n

(

Q2

=

0

)

Previousanalysiscan be extended tothe caseof Q2

=

0 mo-mentswhereabetterconvergenceoftheOPEisexpected [11] and wheretheQCDcontinuumcontributiontothemomentsissmaller aswe shall work with higher moments at which the n-stability isreached. The PTexpression isknown hereup toorder

α

2

s.We

showtheresultsfromthe(ratiosof)momentsinFigs.3and4for

M

n

(Q

2

=

4m2c

)

and

M

n

(Q

2

=

8m2c

)

.We concludethatthe most

stableresults comefromthe momentsfromwhichwe deduceto order

α

2

s (inunitsofMeV):

mc

(

mc

)

|

10_4m2 c

=

1263(1.6)ex

(0.3)

αs

(1.3)

α3 s

(0.2)

G2

(0.3)

G3

(1)

G4

,

mc

(

mc

)

|

16_8m2 c

=

1261(1)ex

(0)

αs

(0.3)

α3 s

(0.1)

G2

(0.1)

G3

(0.8)

G4

.

(9)

ThepreviousresultsarecollectedinTable4. 5. Commentsonmc

(

mc

)

and



α

sG2



from

M

n

(

Q2

)

–Onecan noticeinFig.1that thevaluesofmc

(m

c

)

fromthe

moments

M

n≤2

(

0

)

are strongly affected by the QCD continuum parametrizationthough agree within theerrors withthe onesin [25–28]. Forthecaseofn

=

1 moment used byprevious authors toextracttheirfinalresults,onecandeducefromFig.1:

mc

(

mc

)

|

10

=

1262(59)MeV

,

(10) wheretheerrorisdominatedbythedifferentparametrizationsof theQCDcontinuummodels.Onecancomparethisresultwiththe

one mc

(m

c

)

=

1275

(

23

)

MeV from [25] andthe improved recent

estimate 1279(10) MeV from [26] obtained for

α

s

(M

Z

)

=

0

.

118

from theanalogous n

=

1 moment. The sensitivityof the results onthehighenergypartofthespectralfunctionmayquestionthe accuracyoftheresultsquotedinthesepapersfrom

M

1

(

0

)

.

–Instead,inthen

−

stabilityregion,theQCD continuum-model-dependenceofthe resultdisappears(see Fig. 1) andleads to the optimalandmoreaccuratevaluegiveninEq. (7):

mc

(

mc

)

|

40

=

1266(9)MeV

.

(11) The error due to the parametrization of the spectral function is evenreducedwhenworkingwiththeratioofmoments(seeFig.2) leadingtotheresultinEq. (8):

mc

(

mc

)

|

3₀/4

=

1264(11)MeV

,

(12)

but the errors due to the QCD parameters have increased com-paredtotheoneofthemoment.

–OnecanalsonoticefromtheTablesinRef. [26] thatthe sta-bility of the central values is reached from

M

n=2,3

(

0

)

which is about 10MeV belowtheir favoured choice from

M

1

(

0

)

.A such valueisinabetteragreementwithourpreviousresultsquotedin Eq. (7).

–Weestimate theerrorsinthetruncationofthePTseriesby includingthe

α

4

s contributionassumedto be ofthesamesize as

the

α

3

s one (a geometric growth of the coefficient observed for

massless quarks [29] may not be extrapolatedforheavy quarks). The inducederroris about5MeV whichissmaller thantheone of 19MeV quoted in Ref. [25] estimated usingsome iterativeor contour improvedprocedures wherethe effectof thesubtraction scale

μ

isalsoincluded.

– However, it is not clear that moving the subtraction scale frommc

(m

c

)

tohighervalues,say3GeV [25–28] forimprovingthe

convergenceofthePTseriescanhelpduetotheambiguityofthe charmquarkmassdefinitionsusedintheOPE[

(

1

/m

c

)

expansion].

IndeedaparttheWilsoncoefficientof



α

sG2



knowntoNLO [30],

the ones of the high-dimension condensates are only known to LO. Refs. [26–28] choose towork withthe polemass inthe OPE which, asemphasized in [25] is ambiguousdue to the IR renor-maloncontribution.Then,theuseoftherunningmassintheOPE canbebetterjustifiedwhichisalsoconsistentwiththeuseofthe runningmassinthe PTcontributions. However,ifone movesthe subtractionscale

μ

frommc

(

μ

)

=

1

.

264 to3 GeV, mc

(

μ

)

moves

from 1.264 to 0.972 GeV which can induce an enhancement of about 1

.

3d _{for the dimension d condensate contributions to the}

moments. Therefore,a carefulanalysisincluding radiative correc-tionstotheWilsoncoefficientsofeachcondensateshould bedone when workingat highvaluesof

μ

. Tomy knowledge, thispoint hasnot yetbeencarefullystudied.In ordertocircumvent asuch large enhancement,whichdoesnotarise whenworkingwiththe Laplace sumrule [3] where an optimalvalue of

μ

has been de-rived,welimitheretothe(usualandnatural)choice

μ

=

mc and

donottrytomoveitarbitrarilyaroundthisvalue.

–CoulombiccorrectionshavebeenroughlyestimatedinRef. [1]. However,ithasbeenalsoarguedinRef. [17] thatthiscontribution, whichisnotunderagood control,canbesafelyneglected inthe relativisticsumrules.Therefore,weshallnotconsidersuch correc-tionsinthispaper.

–In [12,17],thesetofQCDparameters:

mc

(

mc

)

=

1275(15)MeV

,

0.7

≤ 

α

sG2

 ×

102

≤

6.3 GeV4

,

(13)

obtainedfromthemoments usedherehasbeen favoured. Exam-iningFigs.4and5of [17],onecanseethatthevaluesofmc

(m

c

)

264 S. Narison / Physics Letters B 784 (2018) 261–265

Fig. 5. Valuesofmb(mb)fromthemomentsMn(0)andtheirratiosrn/n+1(0)and rn/n+2(0)fordifferentvaluesofn usingtheQCDinputparametersinTable1and

Model1givenpreviouslyfortheQCDcontinuumparametrization.Inthen axis:

1≡r1/2,2≡r2/3,3≡r2/4,4≡r3/4,5≡r3/5,6≡r4/5,7≡r4/6,8≡r5/6.

Fig. 6. Valuesofmb(mb)fromthemomentsMn(4mb2)andtheirratiosrn/n+1(4mb2)

andrn/n+2(4m2b)fordifferentvaluesofn usingtheQCDinputparametersinTable1

andModel1givenpreviouslyfortheQCDcontinuumparametrization.Inthen axis:

7≡r7/8,8≡r7/9,8≡r8/9,9≡r8/10,10≡r9/10,11≡r9/11,12≡r10/11,13≡r10/12.

of



α

sG2



due to the absence of mc

(m

c

)

stability versus



α

sG2



.

Thisfeaturehasbeenalsoobservedfromtheanalysisofthesame vectorcharmoniumusingLaplacesumrules [3] whereconstraints from some other charmonium channels are needed for reaching moreaccurateresults.

–Tothevalueof



α

sG2



giveninTable1whichisintheupper

endof the rangein Eq. (13), one can extract from Figs. 4and 5 of [17] thevalue:

mc

(

mc

)

≈

1260 MeV

,

(14)

whichisconsistentwithintheerrors withourprevious resultsin Table4.

– The authors deduce their favorite resultin Eq. (13) from a commonsolutionof themoments andoftheir ratios, whereone cannotice,fromourFigs.3and4,that,atafixedvalueof



α

sG2



,

the value of mc

(m

c

)

from the ratios ofmoments meets the

mo-mentsoutsidethen-stabilityof

M

n

(Q

2

)

,whiletheratiosincrease

rapidlywithn.Thisfactindicatesthatasuchrequirementmaynot bereliable.

–BeyondtheOPE,wecanalsohavesomecontributionsdueto theso-calledDuality Violation,which ismodel-dependent.It can beparametrized (withinournormalization)as [31,32]:

R

D V

c

= (

4

π

2

)

tλve−(δv+γvt)sin(

α

v

+ β

vt

) ,

(15)

where the coefficientsare free parameters and come froma fit-tingprocedure.Foran approximateestimate ofthisadditional ef-fect, we compare its contribution with the QCD continuum one parametrizedbytheasymptoticexpressionofPTspectralfunction

Fig. 7. Valuesofmb(mb)fromthemomentsMn(8mb2)andtheirratiosrn/n+1(8m2b)

andrn/n+2(4m2b)fordifferentvaluesofn usingtheQCD inputparametersin

Ta-ble1andModel1givenpreviouslyfortheQCDcontinuumparametrization.Inthe

n axis:10≡r8/10,11≡r9/10,12≡r9/11,13≡r10/11,14≡r10/1215≡r11/12,16≡ r12/13,16≡r15/17,17≡r12/14,18≡r13/14.

Table 3

MassesandelectronicwidthsoftheϒfamilyfromPDG16[16]. Name Mass [MeV] ϒ→e+e−[keV] ϒ(1S) 9460.30(26) 1.340(18) ϒ(2S) 10023.26(31) 0.612(11) ϒ(3S) 10355.2(5) 0.443(8) ϒ(4S) 10579.4(1.2) 0.272(29) ϒ(10860) 10891(4) 0.31(7) ϒ(11020) 10987(+₋113.4) 0.13(3)

(mc

=

0)(Model 2)fromthethreshold

√

tc=4.5GeV. Weusethe

coefficients:

λ

v

=

0, δv

≈

3.6,

γ

v

≈

0.6,

α

v

≈ −

2.3, βv

≈

4.3

,

(16)

fixed from

τ

-decay data by assuming that they can be applied here. We found that, in theexample n

=

1 and Q2

₌

_0,this ef-fect is completely negligibleeven allowing a low value of

√

tc

=

1

.

65GeV atwhichthefitofthecoefficientshasbeenperformed. 6. Runningmb

(

mb

)

bottomquarkmassfrom

M

n

(

Q2

)

Thepreviousanalysisisextendedtotheb-quarkmass.Weshall use the data input in Table 3. Behaviours of the (ratios of) mo-mentsversusthedegreeofthemomentsaregiveninFigs.5to7. We deduce asoptimalvalues theoverlapping regions of the one fromthemomentsandtheratiosofmoments.Weobtaintoorder

α

3

s (inunitsofMeV):

mb

(

mb

)

|

60

=

4185.9(8.2)ex

(4)

αs

(1.7)

α4

s

(0.8)

G2

(0.2)

G3

(0.2)

G4

,

(17) andtoorder

α

2

s (inunitsofMeV):

mb

(

mb

)

|

10_4m2 b

=

4189.2(6.4)ex

(1.6)

αs

(3.6)

α3 s

(0.5)

G2

(0)

G3

(0)

G4

,

mb

(

mb

)

|

13_8m2 b

=

4187.7(4.3)ex

(1)

αs

(5.0)

α3 s

(0.3)

G2

(0.3)

G3

(0.3)

G4

.

(18) TheseresultsarequotedinTable4.

7. Conclusions

Table 4

Charmandbottomrunningmassesmc,b(mc,b)from

(ra-tiosof)moments.

Observables Mass [MeV] Charm M4_{(0) 1266(9.0)} r3/4_{(0) 1264(11.1)} M10_(4m2 c) 1263(2.3) M16_(8m2 c) 1261(1.3) Mean 1264(6) Bottom M6_(0)⊕r4/5_{(0) 4186(9.3)} M10_(4m2 b)⊕r9/10(4m2b) 4189(7.5) M13 (8m2 b)⊕r10/11(8m2b) 4188(6.7) Mean 4188(8)

useofthehighermomentsandtheirratiosreducenotablythe er-rorsinthemassdeterminations. Thoughitisdifficultto estimate thesystematicerrorsoftheapproach,wecanexpectthattheyare at most equal to the ones quoted in this paper. These new re-sultsarealsoinperfectagreementwiththeonesquotedinEq. (5) from a recent global fit of the (axial-)vector and (pseudo)scalar charmoniumandbottomiumsystemsusingLaplacesumrules [3]. Some comments on the existing estimates of the quark masses and gluon condensates from SVZ-(ratios of) moments are given inSection5.Ourresultsare comparablewithrecentresultsfrom non-relativisticapproaches [33] butmoreaccurate.

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